Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Do Now

7 minutes

As students walk in the room, I hand them a grid with several triangles plotted on it. In a previous lesson on rotational symmetry, students described the rotations that mapped an object onto itself. For this activity, students are given triangles and asked to describe the rotations that map triangles onto other triangles. It leads into the lesson where students will use functions to describe rotations (G.CO.2).

An additional question asks students to explain how a quote relates to transformations. The quote, from Through the Looking Glass, by Lewis Carroll, said by Alice, is about a knight falling off a horse and landing upside-down. I like to use this quote to add some literacy and real-world connections to the lesson. Students recognize the quote as relating to rotations, which turn objects about the plane.

For more information about how and why I used Lewis Carroll’s writing in my classroom, see my video “Lewis Carroll and Mathematics.”

Some students have difficulty identifying the angle and center of rotation. It is helpful to provide these students with tracing paper or patty paper to help them work with rotations.

Rotations Do Now

Mini-Lesson

We begin the mini-lesson by going over the Do Now. First I ask students for their answers to question 2, which is about how the quote relates to rotations. Then I call on students to give their answers to questions 1a, b, and c. This leads into a discussion about various centers of rotations. We discuss rotations about a point in the middle of an object, rotations about a vertex of the object and rotations about a point outside of the object, i.e. the origin. I then have the students label the quadrants of graph from their Do Now. This helps students remember the direction a rotation turns.

At this point, I hand out a sheet for guided notes. We then discuss notation for rotations. Depending on the level of my students, I often hand out tracing paper or patty paper to help with the next part. Students plot three points on each grid and then rotate the triangle about the origin. I find it’s helpful to draw an arrow pointing north on the tracing paper to help identify when the paper has been rotated the varying degrees. After students perform the rotations, they write rules to describe the rotations.

At the end of the mini-lesson, we discuss the effects of rotating an object by a negative degree measure.

Rotations Mini-Lesson

Rotations with Tracing Paper Video.mov

Activity

18 minutes

In this section, students practice rotating triangles and describing rotations. They start out by plotting a triangle on the coordinate plane and rotate it 90o, 180o, and 270o about the origin, which is one of the vertices of the triangle. Then they plot a different triangle and rotate it -90o, -180o, and -270o about the origin, which is not one of its vertices. Students can use their rule sheet from the Mini-Lesson, or they can work without the rules and check their answers using the rules. When rotating an object 90o or 270o, I have the students check that the rotated line is perpendicular to the original line.

Students then answer questions about their rotations. These questions build on the content from the Mini-Lesson and also from previous lessons on transformations.

As the students are working, I circulate around the room. Most of the questions students have can be answered by referring back to the Mini-Lesson or by asking another student.

Because the rules for rotations of different angles are similar, students often make minor mistakes. Even if students don’t use tracing paper to perform their rotations, I give them tracing paper to check their rotations. This helps students check their work and make any adjustments needed.

After about 10 minutes, I stop the students to go over some rotated points to verify they are on the right track. Then they continue working.

Rotations Practice

Summary

5 minutes

To end the lesson, I have students turn and talk to the person next to them. They discuss their answers to questions D and E from the practice worksheet. These two questions have answers that can vary and therefore lead well into a discussion where students have to justify their own answers. After about two minutes of talking in pairs, we discuss as a group. I call on a few students to explain their answers. This discussion leads into the next lesson on mixed transformations where students have to identify the sequence of transformations that maps an object onto another object.